Probabilistic Diophantine Approximation: Randomness in Lattice Point Counting

497 Pages·2014·3.04 MB·English

by  József Beck

Springer Monographs in Mathematics
József Beck
Probabilistic
Diophantine
Approximation
Randomness in Lattice Point Counting
Springer Monographs in Mathematics
Moreinformationaboutthisseriesathttp://www.springer.com/series/3733

József Beck
Probabilistic Diophantine
Approximation
Randomness in Lattice Point Counting
123
JózsefBeck
DepartmentofMathematics
RutgersUniversity
Piscataway,NJ,USA
ISSN1439-7382 ISSN2196-9922(electronic)
ISBN978-3-319-10740-0 ISBN978-3-319-10741-7(eBook)
DOI10.1007/978-3-319-10741-7
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014950069
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Preface
p
Wecouldchooserandomnessof 2asanalternativesubtitleofthebook.Indeed,
p
thebookconnectstwoseeminglyunrelatedconcepts,namely,(1) 2:symbolizing
the class of quadratic irrationals, including the theory of the quadratic number
fields in general and (2) randomness. These two concepts, representing algebra
(thescienceoforderandstructure)andprobabilitytheory(thescienceofdisorder),
are the endpointsof a long chain of relations/implications. The periodicity of the
p
continued fraction of 2 (or any other quadratic irrational) means self-similarity.
Self-similarity leads to independence (e.g., via Markov chains; here we refer to
the well known probabilistic concept), and independence ensures (nearly) perfect
randomness.Inparticular,weprovesomeunexpectedprobabilisticresults:
quadraticirrationalH)periodiccontinuedfractionH)
H)self-similarityH)independence.orindependenceviaMarkovchains/H)
H)randomnessWcentrallimittheoremandthelawoftheiteratedlogarithm
Thisdiagrammaysummarizethebookinanutshell.
p
Thereasonwhywedecidednottochooserandomnessof 2tobethesubtitle
is that it would perhaps mislead the reader. The reader would probably expect us
to prove the apparent randomness of the digit distribution in the usual decimal
expansion
p
2D1:414213562373095048801688724209698078569671875376948::::
Unfortunately, we cannot make any progress with this famous old problem; it
remains open and hopeless (to read more about this and other related famous
open problems the reader may jump ahead right now to Sect.2.5: A Giant Leap
in number theory). What we study instead is the “irrational rotation” by any
v
vi Preface
p
quadratic irrational, say, by 2. We study the global and local behavior of the
irrationalrotationfromaprobabilisticviewpoint—thisexplainsthetitleofthebook
probabilisticdiophantineapproximation.
Consider the linear sequence n˛, n D 1;2;3;:::: it is perfectly regular, it
is an infinite arithmetic progression. Even if we take it modulo one, and ˛ is
an arbitrary (but fixed) irrational, the sequence n˛ (mod 1)—called irrational
rotation—still features a lot of regularities. For example, (1) we have infinitely
manyBoundedErrorIntervals,(2)wehaveinfinitelymanyBoundedErrorInitial
Segments,(3)everyinitialsegmenthasatmostthreedifferent“gaps,”and(4)there
isanextremelystrongrestrictionontheinducedpermutations—theseareallstrong
“anti-randomness”typeregularitypropertiesof theirrationalrotationn˛ (mod1),
n D 1;2;3;:::(properties(1)–(4)willbe explainedin depthsin Sect.1.1).These
regularitiesshowthattheirrationalrotationishighlynon-randominmanyrespects.
Thisiswhytheirrationalrotation(withanunderlyingnestedstructure)isalsocalled
aquasi-periodicsequence.
Also we know from number theory that the key to understand the irrational
rotation n˛ (mod 1), n D 1;2;3;:::; is to know the continued fraction for
˛. The quadratic irrationals have the most regular continued fraction: the class
of quadratic irrationals is characterized by the property of (ultimately) periodic
continuedfraction,forexample,
p 1
2D1C DŒ1I2;2;2;:::(cid:2)DŒ1I2(cid:2):
2C 1
2C(cid:2)(cid:2)(cid:2)
Despitetheseregularitiesoftheirrationalrotation,ourfirstmainresultexhibits
“full-blownrandomness.”Forexample,howmuchtimedoestheirrationalrotation
n˛ (mod 1), n D 1;2;3;:::; spend in the first half Œ0;1=2/ of the unit interval
Œ0;1/?Well,weproveacentrallimittheoremforeveryquadraticirrational˛ (e.g.,
p
˛ D 2). More precisely, let ˛ be an arbitrary real root of a quadratic equation
p
with integer coefficients, say, ˛ D 2. Given any rational number 0 < x < 1
(say,x D 1=2)andanypositiveintegern,wecountthenumberofelementsofthe
sequence˛;2˛;3˛;:::;n˛ modulo1thatfallintothesubintervalŒ0;x(cid:2).Weprove
that this counting number satisfies a central limit theorem in the following sense.
First, we subtractthe “expectednumber"nx fromthe countingnumberand study
the typical fluctuation of this difference as n runs in a long interval 1 (cid:2) n (cid:2) N.
Dependingon˛ andx,wemayneedanextraadditivecorrectionofconstanttimes
logarithm of N; furthermore,what we always need is a multiplicative correction:
divisionby(another)constanttimessquarerootoflogarithmofN.IfN islarge,the
distributionofthisrenormalizedcountingnumber,asnrunsin1(cid:2)n (cid:2)N,isvery
closetothestandardnormaldistribution(bell-shapedcurve),andthecorresponding
error term tends to zero as N tends to infinity. This is one of the main results of
thebook(seeTheorem1.1).Theproofisrathercomplicatedandlong;ithasmany
interesting detours and by-products. For example, the exact determination of the
Preface vii
keyconstantfactors(intheadditiveandmultiplicativenorming),whichdependon
˛andx,requiressurprisinglydeepalgebraictoolssuchasDedekingsums,theclass
numberofquadraticfields,andgeneralizedclassnumberformulas.
p
Perhaps the reader is wondering: why are the quadratic irrationals (like 2)
specialandworthspendinghundredsofpageson.Theansweristhatthequadratic
irrationals play a central role in diophantine approximation for several reasons.
Theyarethe“mostanti-rationalrealnumbers”(officiallycalledbadlyapproximable
numbers), and at the same time they represent the most uniformly distributed
irrationalrotations.AthirdreasonisthePell’sequationx2 (cid:3)dy2 D ˙1(d (cid:4) 2is
p
squarefree),whichisofcoursecloselyrelatedto d.Also,andthisisthemessage
of our book, the best way to understand the local and global randomness of the
irrationalrotation is to focuson the class of quadraticirrationals.This class gives
themostelegantandstrikingresultswiththesimplestproofs.Someoftheseresults
extendtoalmosteveryrealnumber,someofthemdonotextend.Wewillelaborate
oneachoneoftheseissueslater.
The quadratic irrational rotation demonstrates the coexistence of order and
p
randomness; a novelty here is the much smaller norming factor logn (instead
p
of the usual n). The logn comes from the fact that the underlying problem is
about“generalizeddigitsums”withthesurprisingtwistthatthebaseofthenumber
p
systemisanirrationalnumber(namely,thefundamentalunit,e.g.,itis1C 2for
p
˛ D 2).Alsolognrepresentstheminimum;itcorrespondstothemostuniformly
distributedirrationalrotations.
OursecondmainsubjectismotivatedbytheclassicalPell’sequation.Findingthe
integralsolutionsof(say)x2 (cid:3)2y2 D ˙1meanscountinglatticepointsinalong
and narrowtilted hyperbolicregionthatwe call a “hyperbolicneedle.” Of course,
we basically knoweverythingaboutPell’s equation(this is why Pell’s equationis
included in every undergraduate number theory course), but what happens if we
translatethe“hyperbolicneedle”?What istheasymptoticnumberoflattice points
inside (note that the area is infinite)? Well, for a typical translated copy of the
“hyperbolicneedle”—whichcorrespondstoan“inhomogeneousPellinequality”—
weprovea“lawoftheiteratedlogarithm,”whichdescribestheasymptoticnumber
ofintegralsolutionsinastrikinglypreciseway.Inotherwords,theclassicalCircle
Problem of Gauss is wide open, but here we can solve an analogous Hyperbola
Problem. This result is a good illustration of the full power of the probabilistic
viewpoint in number theory. In general, consider the inhomogeneousdiophantine
inequality
c
kn˛(cid:3)ˇk< ; (0.1)
n
where ˛ is an arbitrary irrational, ˇ, c > 0 are arbitrary real numbers, and n is
the variable. An old result of Kronecker states that inequality (0.1) has infinitely
manyintegralsolutionsnifc D3;thisishowKroneckerprovedthattheirrational
viii Preface
rotationn˛(mod1)isdenseintheunitinterval.Whatcanwesayaboutthenumber
p
ofsolutionsnofinequality(0.1)?Considerthespecialcase˛ D 2of(0.1):
p c
kn 2(cid:3)ˇk< ; (0.2)
n
p
andletF. 2IˇIcIN/denotethenumberofintegralsolutionsnofinequality(0.2)
satisfying 1 (cid:2) n (cid:2) N; this counting function is about the local behavior of the
p p
irrationalrotationn 2(mod1).WecandescribethetrueorderofF. 2IˇIcIN/,
as N ! 1, in an extremely precise way for almost every ˇ. We prove that the
p
number of solutions F. 2IˇIcIen/ of (0.2) oscillates between the sharp bounds
(">0)
p p p p p
2cn(cid:3)(cid:3) n .2C"/loglogn<F. 2IˇIcIen/<2cnC(cid:3) n .2C"/loglogn
(0.3)
asn ! 1foralmosteveryˇ;seeTheorem5.6inPart1.3ofthebook.Notethat
p
(cid:3) D(cid:3). 2;c/>0isapositiveconstant,and (0.3)failswith2(cid:3)"insteadof2C".
(Thereasonwhyin(0.3)weswitchedfromN totheexponentiallysparsesequence
p
enisthatthecountingfunctionF. 2IˇIcIN/isslowlychanginginthesensethat,
p
as N runs in en < N < enC1, F. 2IˇIcIN/ makes only an additive constant
change.)
Observethatinequality(0.2)is(basically)equivalenttotheinhomogeneousPell
inequality
(cid:3)c0 (cid:2).xCˇ/2(cid:3)2y2 (cid:2)c0; (0.4)
p
wherec0 D 2 2c. Notice thatequation(0.4)determinesa longandnarrowtilted
hyperbolaregion(“hyperbolicneedle”).Themessageof(0.3)is,roughlyspeaking,
that for almost all translations, the number of lattice points in long and narrow
hyperbolasegmentsof any fixed quadraticirrationalslope equalsthe area plusan
errortermwhichisnevermuchlargerthanthesquarerootofthearea.
Noticethat(0.3)is aperfectanalogofKhinchin’slawoftheiteratedlogarithm
in probability theory (describing the maximum fluctuations of the digit sums of
a typical real number ˇ; the factor loglogn in (0.3) explains the name “iterated
logarithm”).
We also have an analogous central limit theorem: the renormalized counting
function
p
F. 2IˇIcIen/(cid:3)2cn
p ; 0(cid:2)ˇ <1;
(cid:3) n
hasastandardnormallimitdistributionwitherrortermO.n(cid:3)1=4.logn/3/asn!1
p
[(cid:3) D(cid:3). 2;c/>0isthesamepositiveconstantasin(0.3)].
Preface ix
Formally,
ˇ n o
ˇ p p
maxˇmeasure ˇ 2Œ0;1/W F. 2IˇIcIen/(cid:3)2cn(cid:4)(cid:4)(cid:3) n
(cid:4)
Z ˇ
(cid:3)p1 1e(cid:3)u2=2duˇˇˇDO(cid:6)(cid:2)n(cid:3)1=4.logn/3(cid:3); (0.5)
2(cid:5) (cid:4)
wherethemaximumistakenoverall(cid:3)1<(cid:4)<1(andofcoursemeasuremeans
theone-dimensionalLebesguemeasure).
The proofs of the innocent-looking results (0.3) and (0.5) are quite difficult
(in spite of the fact that most of the arguments are “elementary”). Note that
here “independence”comesfrom a goodapproximationby modified Rademacher
functions.
The book is basically “lattice point counting” in disguise. This explains the
subtitle randomness in lattice point counting. The main results are proved by the
samescheme:werepresentanaturallatticepointcountingfunctionintheform
X CX CX C::: C negligible;
1 2 3
whereX ;X ;X ;:::areindependentrandomvariables.Thiswaywecandirectly
1 2 3
applysomeclassicalresultsofprobabilitytheory(suchasthecentrallimittheorem
and the law of the iterated logarithm). We have the following questions: (a) how
toconstructtheindependentrandomvariablesX ;X ;X ;:::,(b)howtocompute
1 2 3
theexpectation,andfinally(c)howtocomputethevariance.Thesearesurprisingly
difficultquestions.
Ofcourse(0.3)and(0.5)extendtoallquadraticirrationals.Theyalsoextendto
some other special numbers for which we know the continued expansion(e.g., e,
p
e2, e).
Someofthemainresultsaboutquadraticirrationals(e.g.,Theorems1.1and1.2)
do not extend to almost every ˛. The reason is that the continued fraction digits
(officiallycalledpartialquotients)ofatypicalrealnumber˛exhibitaveryirregular
behavior(seeSect.6.10).
Someotherresults,including(0.3)and(0.5),dohaveananalogforalmostevery
p p
˛.Thereis,however,adifference:thenormingfactor nisreplacedby nlogn,
andalsotheerrortermismuchweaker(seeSect.6.10).
Thekindof“randomness”weproveinthebookrequiressomeknowledgeabout
the continued fraction expansion of the real number ˛. This is why the best way
to demonstrate this “randomness” is to study the class of quadratic irrationals.
Unfortunately,weknowverylittleaboutthecontinuedfractionofalgebraicnumbers
of degree (cid:4) 3. This explains why we cannot prove anything about (say) the
p
“randomnessof 3 2”;thisiswhywecanprovestrongresultsaboutthe“randomness
ofe,”andcanprovenothingaboutthe“randomnessof(cid:5).”